Bifurcations and limit cycle prediction of rotor systems with fluid-film bearings using center manifold reduction

Many rotating machines utilize a fluid-type bearing. Despite their reliability and high load capacity, these bearings often show instabilities due to the interaction between the fluid media and the rotating shaft. These instabilities, known as oil-whirl and oil-whip, occur due to a Hopf bifurcation; being the parameter the shaft speed. Identifying the type of bifurcation, either sub-critical or super-critical, is an important task to determine the safety of the machines near the instability speed, and it tells whether one experiences oil-whip or oil-whirl. This work presents an approach, based on the center manifold reduction (CMR) method, to obtain limit cycles near Hopf bifurcations of rotors supported on fluid-film bearings. The basis of the method is the obtention of the center manifold of the system, which allows one to assess the type of bifurcation at hand. To obtain the center manifold, the parameterization method for invariant manifolds is used, which is a powerful tool to obtain invariant manifolds of high-dimensional dynamical systems. The proposed method is evaluated by comparing its results with an open-source numerical continuation package (MATCONT) in two systems: a simple and a realistic rotor system. The results show that the CMR, together with the parameterization method, can be used reliably to learn if the rotor system presents sub- or super-critical bifurcations, to perform parametric studies with different bearing properties, and also to predict the amplitude of the limit cycles.